proximities

Proximities refer to the various senses in which things can be near each other. The concept is used across disciplines to describe nearness in physical space, in time, in social relations, or in conceptual similarity. In everyday language, proximity often influences decisions about movement, interaction, and how spaces are organized.

In mathematics, proximity spaces formalize the intuitive idea of nearness without relying on a distance function. A proximity relation on a set relates certain pairs of subsets as being near. Proximity spaces generalize metric spaces and give rise to a natural topology, enabling notions of limit, continuity, and completion. Several standard forms exist, including Smirnov (often called Lodato) proximities and Efremović proximities; each satisfies axioms that encode symmetry, monotonicity, and the preservation of near-ness under unions and supersets.

In geography, urban planning, and the social sciences, proximity describes how closeness in space or social structure affects behavior and organization. Geographic proximity influences accessibility to services and opportunities; social proximity reflects ties within networks or communities. In data analysis and computer science, proximity measures quantify similarity or distance between objects, guiding clustering, nearest-neighbor searches, recommender systems, and network analysis.

Across disciplines, proximate concepts appear in related terms such as proxemics, the study of personal space in human interaction, and in design and policy contexts that consider spatial or relational nearness for planning, communication, and research.